My wife and I have been doing solo gaming since we started dating (way back in the late 80's). 
For the cureent Ed of D&D - if you or the fiance are into character building (in addition to playing) try out the gestalt rules in UA. When we got the book, and I started using them, it helped  - because it allowed the characters to cover more skills.
Once both of you are expereienced at it, go ahead and let the player run two characters. Same as Getstalt - allows for more options, which helps if you want to run a published module.  I run two gestalt characters when my wife GMs me solo, and it pretty much lets me fill all four classic "niches" for trying to get through published modules.
Another thing that is really great about running solo is that you can design adventures around the character and player needs.  Go with the idea that the character(s) are the heroes of legend.  Work that into play, let them be known - personalize the game to them. I've invented an entire game world for no other reason that to let my wife's character get high level and save it*.  - Then the characters really feel like the heroes of The Lord of The Rings.  
Solo play with your significant other can also be ripe with much more roleplaying chances than normal group game, if that is something that you both are into. My wife and I sometimes go multiple sessions with no combat. If the character has a romance, it is also much easier to roleplay in a solo situation, with your significant other**. It is also a great way to just spend time together, which, not worrying about the gaming, is great for the relationship.
My wife added when I mentioned this - if you are an adversairal style GM/player - solo gaming is not a good idea, because it can lead to problems in the relationship.
* And she has done the same for me.
** And without being crass, that romantic roleplaying can be better than a candlelight dinner and romantic music for events later in the evening.